CA1101626A - Polyhedral structures - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
Polyhedral structures which are, for example, self supporting domed buildings having a plurality of faces arranged in a symmetric pattern, and in a preferred embodiment, the gridwork structure is defined by sixty identical equilateral triangular faces; having two parallel hexagonal sides of six triangles each that are bounded by six side sections each consisting of eight triangles forming a pair of pentagonal pyramids, that have two triangu-iar faces in common. This equilateral faced gridwork is expandable without bound to form larger polyhedra domed buildings by systematically adding groups of additional face structures. Numerous series of Face edges con-necting end to end, circumscribe the polyhedra domed building to serve as truncations to form juncture walls or a base on which the dome portions of the structure can be supported.

Description

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BACKGROUND OF THE INVENTION
Since the invention of the geodesic dome by R Buckminster Fuller in 1951, the use of domed enclosures has become widespread and there is now extensive technical and popular literature in the field. The primary engineerin~ advantage of a domed bui Iding enclosure is that a sphere enclosed the most volume with the least surface area of any geometrical shape and the sphere is the strongest shape against internal and radial pressure.
There are also other advantages of these sphere like dome buildlng enclo-sures; a dome is essentially self supporting; all structural members are 10 adjacent to or part of the faces or the skin of the dome; and the triangular pattern of the faces or the skin grid results in desirable load distribution and rigidity of the overal I structure.
The term geodesic relates to a method of subdividing the surface of a sphere into a triangular grid pattern of intersecting great circle arcs and utilizing struts which are chords of these arcs to form the skeletal -Fu framework for a generally spherical enclosure. The resulting structure is actually a polyhedron, a multi-sided, spherically symmetrical figure which more closely approximates a sphere, as the frequency of subdivision in-creases. If the polyhedron is to have faces which are all identical, equi-20 lateral triangles, the largest figure which can be constructed is the icosa-hedron7 a twenty sided shape which has twelve identical corners, each of which is the apex of a pentagon, and thirty edges, all of equal length.
Mathematicians have also proved that each of the twenty equi lateral icosa-hedron triangles can be further divided into six identical triangles having 30-60-90 angles and that this is the largest number of identical triangles into which a sphere can be subdivided. This type oF geodesic dome is known as a two frequency icosahedron and has 120 identical faces. Greater subdivision results in higher frequency figures which more closely approxi-mate a sphere but also require a larger number of strut lengths and a 30 variety of different apex connectors.

Although a geodesic dome structure has many advantages over conventional building structures, it also has disadvantages. A hemispherical dome has a large amount of space near its perimeter, which has only limited use, because the dome arches toward the ground resulting in a very low :

~` ceiling~ Also as the size of the enclosure increases, greater frequency of subdivision results in an increasing number of non-interchangeable parts.
Moreover, it is difficult to divide or add onto the basic hemispherical shape of a geodesic dome structure.
Although there are an infinite number of multi-sided building constructions which can be assembled from identical panels, the basic struc-ture of the present invention, has sixty identical equilateral triangular faces.
This basic polyhedral structure was invented because its shape results in less unusable enclosed space than does a sphere and because it is also easily 10 expanded or subdivided, while retaining a great deal of rigidity and strength.
The symmetry properties of this basic polyhedron structure can be expanded upon to generate a variety of domed structures which have these advantages and additional advantages further described herein. These expanded poly-hedral domed structures are systematically designed and constructed, so the number of new components, such as faces and struts, which are employed are kept to a minimum.
SUMMARY OF THE INVENTION
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These polyhedral dome~l building structures are designed to have many advantages over conventional geodesic or other geometric domed bui Id-20 ing structures. A family of unique polyhedrons defines the many overalldomed shapes of the many illustrated and possible embodiments of these poly-hedral structures. [n all dwelling embodiments, there is an orientation of struts and their connecting hubs to form a dome gridwork, which ultimately corresponds to the edges and vertices of a plurality of panels which collecti-vely form the faces of the polyhedra structure.
These polyhedra dome structures, after assembly, are rigid and self supporting, can be oriented to be built with substantially vertical side ` walls, creating a great deal of usable space within these dome structures which will accomodate, for example, conventional furniture. The polyhedron 30 family of embodiments of these dwelling structures is also selected to maxi-mize usage of identical, interchangeable parts and also to be easily construct-ed with economical building materials.
The basic polyhedron domed bui lding structure from which al l `- embodiments are derived, has 90 equal lengthed edges meeting at 32 vertices to form 60 equilateral triangular faces. This basic embodiment is derived during actual construction by first creating two parallel hexagonal sides~
each formed with six triangular faces, which are thereafter ioined at their edges~ by six identical side sections, each consisting of a pair of pentagonal pyramids which have two triangular faces in common. A hexagonal prism -~
has six planes of reflectional symmetry which share a common ax7s passing through the center of the two planar hexagons and a seventh plane of symmetry which is perpendicular to this axis and bisects the prism. This polyhedron retains these planes of symmetry. Further, this polyhedron with 10 faces of equilateral triangles can exist in more than one form that retains this symmetry, and many forms that have other symmetry patterns. The form -shown in Figure 1 is chosen because it encloses the most volume ancl has the best structural pattern.
A systematic method of varying the strut lengths, further descri-bed and illustrated herein, may be accomplished in a way that preserves the symmetry of the basic shape, and hence preserves also the maximum number of identical parts. The vertex pattern arrangement thus formed can have curvatures such as parabolic or geodesic.
A method of expansion of the basic polyhedron is accomplished by -~
folding the faces into three planar sections and then adding groups of addi-tional faces and refolding the faces to form enlarged polyhedra. The poly- ~ '~
hedron can be expanded indefinitely while adding parts which are nearly identical and interchangeable. Conventional geodesic expansion, on the other hand, would requi re an increasing number of strut lengths and form many different triangular faces as the frequency of the subdivision increases.
Another advantage of the polyhedron shapes used in this building structure is that subdivision is possible along any one of several series of connected struts to leave a substantially planar face which may be a base on which the truncated structure can be supported. The planar truncation could 30 also be usedas a side wall or as a junction for connection to another similar ; enclosure or to a conventional rectangular building. At least nine substan-tially planar truncations of the basic polyhedron are defined by series of struts which are connected end to end and completely circumscribe the poly-hedron.

_3--~6~6 Rigid concave-convex hubs having curvature which tends to round off the corners of the polyhedron and smooth load paths, may be used at each vertex of the building structure to connect struts in the required angular al ignment.
DESCR!PTION OF PREFERRED EMBODIMENTS
Basic Embodiments The building structures described herein are derived from the polyhedron 10 shown in Figures l, 2, 3 and 4 which is also a preferred embodiment of the invention. This polyhedron is constructed with sixty 10 identical, equilateral triangular faces 14, defined by ninety equal lengthened struts 12. The polyhedron 10 has two parallel, planar, hexagonal sides 30, 32 each formed by six triangular faces 14. Although one advantage of this structure is that many orientations of the polyhedron building framework are possible, the alignment shown in Figure 1 will be referred to herein and the - two planar hexagons designated as the top 30 and base 32 hexagons.
This basic polyhedron 10 has a hexagonal top 30 and base 32, connected by six segments, each consisting of eight equilateral, triangular faces forming a pair of pentagonal pyramids 18 which have two faces in common. Each of twelve pentagonal pyramids 18 in the polyhedron 10, has one edge in common with an edge of either, the top 30 or base 32 planar hexagon. The six pentagonal pyramids 18, which have edges in common with and depend from the top planar hexagon 30 are paired with the corresponding six pentagonal pyramids 18 which have edges in common with the bottom planar hexagon 32, such that each pentagonal pyramid 18 shares two of its five triangular faces with its paired pentagonal pyramid 18.
These shared triangular Faces From the circumferential side band 46 of the dome that is perpendicular to the two hexagons forming the top 30 and base 32 of the polyhedron 10, thereby forming a vertical sidewall.
Subdivision and Truncation of the Basic Embodiment This polyhedron dome 10 may be readily subdivided along any one of numerous series of connected edges or struts ~8 which ci rcumscribe the polyhedron and lie substantially within a plane. Hexagonal apex 16 is the intersection of three such series of co-planar edges. Each edge 26 of the planar hexagon 307 shown in Figure 2 is part of three series of substantially 6~6 ; , planar co-terminous struts 48 that are almost copianar, and many series that haveggreater variation from a co-pianar condition. Subdivislons of the polyhedron 10 along any of these series of struts 48, leaves a surface which can be a base or sidewall ~r the junction with another structure. Additional planes of truncation which require faces 14 to be subdivided are also possible.
Variations of the Basic Embodiment While the equilateral properties of the dome 10 minimize the number of different components, there are other advantages that result when other 10 triangle types are used. For example, the two planar hexagonal apexes 16 form the weakest parts of the structure as six struts intersect, all Iying wjthin a plane. These struts may be increased in length to result in tri-angulation of the forces at a peaked hexagonal apex, greatly increasing the load bearing strength of the now non-planar top 30 of the polyhedron.
Another consideration is the fact that covering materials are commonly manu-factured in four by eight foot rectangles. These rectangles can be cut dia-gonally to cover an isosceles triangle without waste, when the sides of the triangle are chosen suitably. -The important criteria in considering variations of basic dome 10 -~
20 is that many identical interchangeable parts result in symmetry of the polyhed-ron. Conversely, if a polyhedron is constructed from a smal I number of different parts by the operations of rotation about an axis and reflection in a plane then it will be symmetric and have a large number of identical parts.
The dotted lines in Figure 2 show where six planes 71, 72 bisect the poly- ~`
hedron 10, a theoretical line passing through the two points 16, being their common intersection and axis. ,~. seventh plane 73 is perpendicular to this axis and also bisects the polyhedron as shown in Figures 3 and 4. The whole polyhedron may be generated by reflecting the four triangles 80, 81, 82 and 83 in these seven planes. In Figure 31 these four triangles are shown 30 as a flat planar net, the dotted lines indicating how they are cut the planes of reflection symmetry 71 and 73. In Figure 32 this planar net is shown in a modified form where triangles 80, 81, 82, 83 are iscoscles. This modi Fied net can be refolded in space so that the vertices 22 and 24 lie in the planes `~ 72 and the vertices 20 lie in the planes 71. The four triangles thus oriented : `

' can be reflected through the seven planes of symmetry to produce a varia-tion which preserves the symmetry of the basis embodiment 10. ~ Variation of this kind is called a "symmetr7c embodiment" oF the invention and is dep i ct ed i n F i gu re 33 .
IF triangles 80, 81, 82 and 83 are made congruent isosceles triangles, when this net is foled and reflected as described above, the resulting polyhedron is called a 'Isymmetric isosceles embodiment" of the invention. ~t is made from ldentical isosceles triangles.
Planes of symmetry can be selectively removed to produce further 10 modifications which contain less identical parts. For example, the horizontal plane 73 can be removed to produce "vertically symmetric embodiments".
Since the halves above and below ~he horizontal plane 73 have different triangles and different nets, symmetry in this plane has been sacrificed.
These examples show how a variety of embodiments may be con-structed with different types of symmetry, the number of identical parts decreasing with the successive removal of the planes of symmetry. It is possible to construct embodiments with no symmetry at all, termed "asymme-tric embodiments".
All of the var:ations considered above come under the following 20 definition.
A Comprehensive l:~escription of Stairls Basic Polyhedron "A polyhedron with sixty triangular faces, ninety edges, and thirty two vert ices. "
"The thirty-two certices are of four kinds:
1. Twelve vertices ~here five edges meet, cal led 'pentagonal verticesl.
(Identified in Figure 1 as 20~ )

2. Twenty vertices where six edges meet, cal led 'hexagonal vertices ' .
30 The twenty hexagonal vertices are of three kinds:
; 3. Two vertices having a common edge with six other hexagonal vertices, called Icap vertices'. (Identified in Figure 1 as 16.) 4. Twelve vertices having an edge in common with two pentagonal vertices and an edge in common with Four hexagonal vertices, .

called ~meridian vertices~. (Identified in Figure 1 as 22,)) 5. Six vertices having an edge in common with two hexagonal vertices and four pentagonal vertices, called lequatorial vertices'. (Identified in Figure 1 as 2~. )"
Systemat c_E~_nsion of _he Basic Polyhedron Figures 5 through 8 i I lustrate a method by which the siY~ty faced polyhedron 10 may be systematical Iy expanded to form larger bui Iding struc-tures. ~n Figure 5, x, y and z axes are designated to be used as a frame of reference for such expansions The z axis passes through the two planar `
10 hexagonal vertices 16 and is perpendicular to the two planar hexagons 30, 32.
The x axis is oriented parallel to any strut which is a spoke of the planar top hexagon 30. In Figure 6 the polyhedron 10 is shown folded into three sections. A circumferentia' band of triangles 46 surrounds the polyhedron 10, comprising the shared faces of the six pairs of pentagonal pyramids 18, and forming a vertical side wall of the polyhedron 10. This band of tri-angular faces 46 is shown unFolded into a flat pattern in Figure 1. Two identical flat gridwork patterns 56, 58 are formed by folding the remaining triangular faces into the planes of the two planar hexagons 30, 32, as illustrated in Figure 6. As shown in Figure 7, the flat gridwork pattern 20 S6 is hexagonal in shape and includes the original planar hexagon 30 plus an additional arcade or band of eighteen triangular faces. This band of eighteen triangular faces along with the twelve triangular faces in the cir-cumferential band 46, when assembled in the polyhedron 10 of Figure 5, form the thirty faces of six pentagonal pyramids 18.
With the polyhedron 10 illustrated in Figures 1 through 5 dis~
mantled into two flat gridwork patterns 56, 58 as shown in Figure 7, plus the band of triangular faces 46 shown unfolded in Figure 11, it is possible to add identical triangular faces to expand the structure whi le retaining symmetry of the dome shape.
30 Expansion Along theX and Y and Z Axes Figure 8 i I lustrates the addition of two incremental bands 63 and 65 of eight triangles each expanding the flat pattern 56 in the x direction.
When both the top 56 and bottom 58 flat gridwork patterns are expanded in this way, a lengthened circumferential band 96 as shown in Figure 16 is used ., . ~

which includes four square panels 54 and twelve triangular panels 14. Thus an expansion by two increments in the x direction would result in a domed building structure having ninety-two equilateral triangular faces and four square faces.
Expansion of the gridwork flat patterns 56, 58, Is also possible in the y direction as shown in Figure 9, where expansion of two increments 67, 68 is illustrated. Figure 10 combines the expansion in the x direction as shown in Figure 8 and the y direction expansion shown in Figure 9.
\I\lhenever the domed building structure is expanded in either the x or y 10 direction, faces must also be added to the circumferential band of panels as shown in Figures 14, 15 and 16. Figure 14 illustrates the circumferen-tial band 92 of sixteen equilateral triangles 14 which is required in a dome having an expansion of one increment in the y direction.
Expansions in the z direction are illustrated in Figures 12, 13, 17 and 18, where'the circumferential band 46 is enlarged by the addition of faces to widen it.
Thus three distinct directions of expansion are possible as are combinations of such expansions. The breakdown of the dome structure into three flat components 46, 56, 58 provides a convenient starting point from 20 which systematic, symmetrical expansions may be made, enlarging the poly hedron indefinitely.
Examples of Expanded Domes In Figure 20 a dome 14 having seventy-eight faces is shown which is derived from the sixty sided polyhedron 10 by expansion by one increment in the x direction. ~ lengthened circumferential side 94, illustrated in Figure 15, is required which has two square faces 54 and twelve triangular faces 14. The shape of this expanded polyhedron 15 is more complicated than the polyhedron 10 shown in Figure 1, however, a gridwork oF equal lengthened struts 12 is still formed and identical, equilateral Faces 14 are 30 defined with the exception of the two square faces 54 as shown in Figures 15 and 20. Conventionally shaped windows or doorways can be framed within these square faces 54. This expanded dome building structure 15 can be constructed from two identical flat patterns 56, 58 as illustrated in Figure 8 and the extended circumferential side 94 shown in Figure 15.

-B-The polyhedron 10 expanded by two increments in the z direction is illustrated in Figure 21. Fiyure 7 illustrates the flat gridwork pattern oF the hexagonal top 30 and bottom 32 used to construct this dome. A cir-cumferential band of panels 90, expanded by two increments in the z direc-tion used in this dome is shown folded flat in Figure 13. Thus two grid-work panels as shown in Figure 7 and one as shown in Figure 13 can be `
assembled to form the dome illustrated in Figure 21, having eighty-Four equilateral triangular faces.
A third embodiment of an expanded dome structure is illustrated in Figures 22, 23, 24, and 25. This dome 1~9 is derived from the basic polyhedron 10 shown in Figure 5 by expanding one increment in the x direc-tion, one increment in the y direction and two increments in the z direction.
The circumferential side panel 88 is shown folfed flat in Figure 18.
Variations of Expansion As with the basic embodiment, variations of the strut lengths can be used to satisfy specific application needs such as: alter load distribution characteristics; accommodate VariOUS material sheet sizes; and/or insure water runoff.
Modu I ar Extensions The remarkable structural property of the present invention, which is shared also by other triangulated structures such as the geodesic dome, is that it will hold its shape even with flexible vertex connectors if the struts are rigid. A polyhedron which does not have this property is the rectangular prism, which is the basic geometric shape on which the over-whelming majority of modern buildings are s~ructured. A major reason for the widespread appeal of such a non-structural form is the ease with which a rectangular prism may be subdivided into units which are multiples of the whole, as well as being able to be extended in the same way, combined with the ability of such a simple modular system to adapt itself to mass produc-tion. The embodiments of the invention share this modular expansion property of the rectangular prism while retaining the considerable structural advan-tages of the geodesic dome. Polyhedral units consisting of standard identical triangles and their modular subdivisions can be fastened to basic embodiments of the invention to create structures with greater numbers oF equal parts that _9_ ., ,, ~ , 72~

also have m3re vertical wall space and in fact are even closer approximations to rectangular prisms, but still without their structural disadvantages.
In Figure 33,~` a variation of the basic embod7ment is shown. It is constructed from two different isosceles triangles 92, 93. In Figure 34, two possible rhombic base pyramids 95, 97 are constructed so that their faces are formed from triangles 92, in the case of pyramid 97, and pyramid 95 is constructed from triangles 92 and 99 where 99 is obtained from 92 by bi-section. These pyramids can be attached to the structure in Figure 33 to produce a structure as shown in Figure 35 which consists only of triangles 92 and their subdivision 99. This structure has much more vertical wall space and usable floor area than the em~odiment shown in Figure 33. It may be covered with 30 uncut sheets of ~ x 8 foot material and 21 sheets cut on the diagonal. Similarly, modular additions can be added to any embodiment discussed herein.
DESCRIPTION OF DR~WINGS ~ ;
Figure 1 illustrates the gridwork of the basic polyhedron domed bui lding structure 10;
Figure 2 is a top view of Figure 1;
Figures 3 and 4 are side views of Figure 1;
Figure 5 is a perspective view of the basic domed bui Iding struc- ;i ture, with x, y and z axes designated as a frame of reference;
Figure 6 illustrates the domed building structure folded into two planar flat patterns, leaving a circumferentia! band of faces;
Figure 7 is a top view of the flat pattern shown in Figure 6; ~`
In Figure 87 the flat pattern has been expanded by two increments in the x direction;
Figure 9 illustrates a planar flat pattern expanded by two incre-ments in the y direction;
In Figure 10, a flat pattern is shown expanded by two increments in the x direction and two increments in the y di rection;
In Figure 11, the circumferential band of faces of Figure 6 is shown severed and folded flat;
Figure 12 illustrates a circumferent7al band 69 of panels, used in a dome which has been expanded by one increment in the z di rection;

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Figure 13 illustrates a circumferential band of panels 90 used in a dome which has been expanded by two increments in the z di rection;
Figure 14 illustrates the extended circumferential band g1 of panels for a dome which has been expanded by one increment in the y direc-tion;
In Figure 15, a circumferential band of panels is shown for a dome which has been expanded by one increment in the x direction;
Fi~3ure 16 illustrates a circumferential band 96 after a second :
incremental expansion in the x direction;
Figure 17 shows a widened ci rcumferential band 98 of panels ~ ~:
expanded by two increments in the z direction and one increment in the x .
direction; .
The circumferential band of panels shown in Figure 18 is used in a dome which has been expanded once in the x direction, once in the y direction and twice in the z direction;
Figure 19 shows how a rectangular sheet of material may be cut to make an equilateral triangle. If the sheet is a 4 x 8 foot sheet, the resulting waste 74 is approximately one percent;
Figure 20 illustrates a dome gridwork which has been expanded ` 20 by one increment in the x direction;
:~ Figure 21 illustrates a dome gridwork which has been expanded by two increm2nts in the z direction;
Figure 22 illustrates a dome gridwork which has been expanded by one increment in the x direction, one increment in the y direction and two increments in the z direction;
Figure 23 illustrates a top view of the dome shown in Figure 22;
Figures 24 and 25 illustrate side views of the dome shown in Figure 22;
Figure 26 illustrates a hub connecting five struts at a pentagonal 30 vertex in the dome gridwork;
Figure 27 is a Partial cross sectional view of a double hub assemb l y;

Figure 28 is a view of the interior side of a hub;
Figure 29 is a cross sectional view of a single hub assembly;

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Figure 30 is a perspective view of a hub 36 used at a hexagonal vertex in the dome gridwork;
Figure 31 shows the four generating triangles as a planar net;
Figure 32 shows a modified form of the net shown in Figure 31;
Figure 33 shows a VariatiOn of the basic embodiment constructed from two different 7sosceles triangles;
Figure 34 shows two rhombic pyramids constructed so that the four base edges will coincide with four edges of Figure 33; and Figure 35 shows one way the rhombic pyramids of Figure 34 may be attached to the polyhedron of Figure 33.
LOAD TF;~ANSMITTING COUPLINGS
ln Figure 26, a perspective view illustrates a one piece hub 38 used to connect struts 12 of these bui Idings structures at a pentagonal ver-tex 20. A second, inner hub 76 may be connected to the inside of each vertex of the building structure as illustrated in Figure 27 to provide a particularly strong structure having improved sealing and insulating proper-ties. A more economic and lightweight dome gridwork utilizes single outer hubs 38 as shown in Figure 29. In either the single or double hub configura-tions, nuts 60 and bolts 62 are used to fasten the hubs and strut ends. The hub 38 i I lustrated in Figure 26 through 29 has strut receiving slots 44, 49 designed to receive tubular struts 64 or standard framing lumber 66. Strut receiving slots 49 are provided to secure additional struts which may be used to subdivide dome faces 14.
For each type of vertex required for a particular dome, a hub is designed with the proper geometrical alignment of strut receiving slots and radius of curvature to connect the gridwork of struts. Four different hub configurations are used in the sixty faced dome 10. A larger number of different hub shapes are required for expanded dome confi~3urations. In any of its embodiments, the cast or molded one piece hub 38 has a concave-con-vex shape to integrate the hubs with the strut gridwork, resulting in smooth overal I curvature and even load distribution. The convex outer surface 40 of the hub also provides a continuous interface between triangular panels or other covering skin 59 of the dome, as i I lustrated in Figures 27 and 29. ~
sealant material 70 is shown at the hub-skin interface and also covering bolts ! . ,`

62 which are exposed to the dome exterjor. `
SUMMAF~Y_OF ADVANTAGE5 The dome space enclosing structure which is described herein has many advantages over conventional rectangular build7ngs and also has features which are improvements over geodesic or purely spherical dome structures.
1. This building structure is not purely spherical and several possible orientations of the structure are possible.
2. A high degree of symmetry allows a plurality of identical ~ !
lightweight components, resulting in a building structure that may be easily 10 and qu i ck I y assemb I ed or di sm ant I ed .

3. By symmetrically altering strut lengths, the vertex pattern can have parabolic curvature, which contributes to a large ground area ~`
being covered with a small an~unt of structural weight.

4. The triangulation oF the skin gridwork of the structure results in a frame rigidity and good load distribution.

5. A greater amount of usable space is enclosed than under a purely spherical dome, particularly around the periphery of the enclosed a rea .

6. The basic polyhedron building structure may be systematically 20 expanded as additional faces are added

7. Numerous identical, interchangeable Parts are used in this building structure which may be assembled to form many embodiments with all struts of equal lengths and with equilateral triangular face panels.

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8. The basic polyhedron shape has numerous series of connected ~ ~ `
edges through which substantial Iy planar truncations may be made, and as the polyhedron is expanded the number of different orientations and truncations increases resulting in architectural flexibility.

-9. These domes are more compatible with conventional building materials and methods than is a hemispherical dome.

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Claims (4)

The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows:

1. A polyhedral structure having top, bottom and side elements, comprising:
(a) the top element comprises six triangular faces forming a planar hexagon and a pluraity of triangular faces forming the periphery of said top element, with six of said peripheral faces each sharing one edge with the respective six planar triangular faces and the peripheral faces being inclined downwardly from the planar hexagon;
(b) the bottom element comprises six triangular faces forming a planar hexagon and a plurality of triangular faces forming the periphery of said bottom element with six of said peripheral faces each sharing one edge with the respective six planar triangular faces and the peripheral faces being inclined upwardly from the planar hexagon;
(c) the side element comprising twelve triangular faces spanning and connecting the peripheral faces of said top and bottom elements, said twelve side faces being substantially perpendicular to both the top and bottom planar hexagons.

2. A polyhedron structure as claimed in claim 1, wherein:
the side element and the peripheral faces of the top elements combine to form six pentangonal pyramids and similarly the side elements in combination with the peripheral faces of the bottom element forms six pen-tagonal pyramids.

3. A polyhedron as claimed in claim 2, wherein:
each face of the polyhedron is an identical triangle.

4. A polyhedron as claimed in claim 2, wherein:
each face of the polyhedron is an identical, equilateral triangle.